In the beginning was the circle, not the square. The square came later. This is what John A. Parker seems to be saying in his chapter two proposition 3 from his book, The Quadrature of the Circle (1874 edition).
Proposition 2-3: “The circle is the natural basis or beginning of all area, and the square being made so in mathematical science, is artificial and arbitrary.”
In Parker’s Proposition 2-1 (chapter 2, proposition 1), he had shown in part that all shapes formed of straight lines and equal sides have their areas equal to half the circumference multiplied by the least radius which the shape contains. (Background: “Curved Lines Are Different”, Ersjdamoo’s Blog entry of August 14, 2013). This applies to all regular shapes: their areas are equal to half the circumference multiplied by the radius of an inscribed circle, which is the least radius the shape contains. This is depicted in Parker’s Plate 9, which hopefully appears above. (By clicking on the image, it should enlarge for easier viewing.)
Because of all regular shapes having their areas equal to half the circumference (or perimeter) of same multiplied by the radius of an inscribed circle (i.e., the least radius), “the circle is therefore the basis of area in all such shapes.”
In Parker’s Plate 9 (above) it is shown, for example, how a triangle and a square each have a greater and a lesser radius, and that a circle based on the least radius is less than the shape, i.e., it is an inscribed circle.
Furthermore, neither of the example shapes (triangle and square) can be so diminished that a circle based on its least radius would not still be inscribed within the shape and therefore have an area less than the shape.
“Therefore if either shape shall be diminished to infinity, so long as it shall have magnitude or area, it shall also have a greater and a lesser radius, and a circle described upon its least radius will be less than it.”
Therefore the circle is the least of all possible magnitudes.
Because all extension must be from the least possible magnitude to that which is greater, therefore the circle is the beginning of plane extension.
In the beginning was the circle.